Simplify and expand the following expression: $ \dfrac{4n}{4n + 5}+\dfrac{4n + 9}{3n - 3} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(4n + 5)(3n - 3)$ Multiply the first term by $\dfrac{3n - 3}{3n - 3}$ $ \begin{align*} \dfrac{4n}{4n + 5} \times \dfrac{3n - 3}{3n - 3} & = \dfrac{(4n)(3n - 3)}{(4n + 5)(3n - 3)} \\ & = \dfrac{12n^2 - 12n}{(4n + 5)(3n - 3)}\end{align*} $ Multiply the second term by $\dfrac{4n + 5}{4n + 5}$ $ \begin{align*} \dfrac{4n + 9}{3n - 3} \times \dfrac{4n + 5}{4n + 5} & = \dfrac{(4n + 9)(4n + 5)}{(3n - 3)(4n + 5)} \\ & = \dfrac{16n^2 + 56n + 45}{(3n - 3)(4n + 5)}\end{align*} $ Now we have: $ = \dfrac{12n^2 - 12n}{(4n + 5)(3n - 3)} + \dfrac{16n^2 + 56n + 45}{(3n - 3)(4n + 5)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{12n^2 - 12n + 16n^2 + 56n + 45}{(4n + 5)(3n - 3)} $ $ = \dfrac{28n^2 + 44n + 45}{(4n + 5)(3n - 3)}$ Expand the denominator: $ = \dfrac{28n^2 + 44n + 45}{12n^2 + 3n - 15}$